All this is available elsewhere, but as a trusted source, I'll oblige an election night party request for a math-with-murph explaining the electoral college, the short form is that the EC exists because the Constitution says so--Article 2, Section 2, Clause 2:

Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, equal to the whole Number of Senators and Representatives to which the State may be entitled in the Congress: but no Senator or Representative, or Person holding an Office of Trust or Profit under the United States, shall be appointed an Elector.

And in the 12th Amendment:

The Electors shall meet in their respective states, and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same state with themselves; they shall name in their ballots the person voted for as President, and in distinct ballots the person voted for as Vice-President...

As with many aspects of the Constitution, this is an artifact of the United States of America being set up as, well, United States, rather than United People. Under the Constitution, the people don't elect the President: the States' Electors elect the President. The Constitution leaves it up to each State to determine how to appoint Electors. In theory, Michigan could decide that the Governor gets to pick its Electors. Or that our State Senate gets to pick our Electors. Or that our Federal Senators and Representatives are themselves our ElectorsEdit: Okay, that one's explicitly forbidden. Momentary reading comprehension fail. ...But we potentially could in theory have each Federal legislator individually appoint an Elector, at least until the courts said that was too close to the same thing.

In reality, we've become so accustomed to the idea that we the people get to elect the President that every State uses the vote within the State to determine that State's Electors. (Maine and Nebraska are slightly different: they each have 2 Electors selected by state-wide popular vote, and the other two selected by the popular vote within Congressional district boundaries, so they can have 3:1 Electoral vote splits within the state.)

This raises a few potential issues. One is that an individual person's vote is worth different amounts in different states, due to the "equal to the total of Senators and Representatives" thing--the small states have more Electors per capita than the large States, because of that 2 Senators per State thing. Assuming all eligible voters vote, a Wyoming resident's vote is worth about 1/200,000th of an Electoral College vote, while a Texan's vote is worth 1/750,000th of an Electoral College vote. If you're especially concerned with "1 person, 1 vote", this probably bothers you. This graph on Wikipedia shows the votes per Elector breakdown:

This mismatch of Electors to individual voters is also why the national popular vote can go one way, but the EC vote go the other, or why the popular vote can be so close, while the EC is a blow-out. We all remember 1980 as an absolute domination by Reagan, with 90% of the Electoral College vote--but in the popular vote, Reagan only got 50.7%, and only beat Carter by 10% of the popular vote. In that election, the image of a nationwide Reagan blowout at the EC level probably served to create a "mandate" that in reality was a lot more slim.

The other is that the "winner take all" system of the State-level votes means that candidates have no reason to campaign in "safe" states: when was the last time you heard about a Presidential candidate campaigning in California, or Texas? Or even Wyoming, which, according to the per-capita EC distribution, should be really important? The national popular vote just isn't important in electing the President, so the candidates wage state-level campaigns in a couple of states that "matter", because they are close to the 50/50 margin.

Again, you might think it slightly unfair that certain states never get presidential candidates making commitments towards their interests. Until a month ago, when had you ever heard a candidate talk about the importance of protecting NJ and NY from catastrophic storms? ...Or about anything else specific to the northeastern seaboard? Pretty much never. Democratic candidates know they can count on these EC votes, and don't need to work for them. Republican candidates know they will never ever get these votes, so can safely dismiss them. A popular vote-based presidential election, by contrast, would mean that the whole country's concerns matter, and not just those in a handful of "swing" states.

Obviously, we're stuck with the Electoral College, because it's written into the Constitution. Several states, though, have signed on to a movement to work around the EC, though, through the National Popular Vote movement: since each state has the ability to assign electors however they want, several states have committed through legislation that, they will assign their Electoral College votes to whomever wins the national popular vote, regardless of the vote within that State, once enough States have made the same commitment that they have enough total Electoral College votes to determine the winner. This legislation is all designed to sit dormant until 270 EC votes worth of states pass such laws, at which point it kicks in and creates a de facto popular election for President. So far, 9 states, totaling 132 EC votes, have signed on. Tellingly, none of them are "battleground" states. (Legislation passed the Michigan House in 2009, but didn't pass the State Senate.)

Generally, I support this effort. I don't think it would directly change the results of elections, as only 3 elections in history have seen the two numbers split: popular vote losers Hayes (v. Tilden) in 1876, Harrison (v. Cleveland) in 1888, and Bush (v. Gore) in 2000 all won the EC vote. (Perhaps notably, in all of those cases the Republican candidate came out on the winning side of the EC.)

Much more likely, a popular vote election would smooth out the "spikiness" of the EC vote--like that thing about in which Reagan only got 50.7% of the popular vote in 1980--hopefully illustrating our national purpliness and reducing talk of artificial "mandates" created by EC "landslides". Dichotomous red/blue maps of states would all but disappear once we got away from State-level winner-take-all EC selection--though I do have some concern the media would just move to county-level red/blue maps as a dumbed-down way of representing the popular vote.

We'd also see the campaigns, the media, and the spending spread across much more of the country. We'd probably hear a lot more about immigration, for example, from both sides, once California and Texas become fair game for campaigning. With the "big city" states thrown into the campaign--New York, Illinois, California, Texas, Pennsylvania, and Arizona are home to the country's ten largest cities, and only PA a current "battleground"--we might actually hear some candidate outline an urban agenda. Yet various sources have noted that neither candidate mentioned "cities", or related words or issues like "urban", "transit", "poverty", etc., during their convention speeches, their debates, or most of their campaign speeches. For a majority-urban nation, that's lamentable.

Another thing I think it wouldn't do is affect turnout that much. While broader campaigning (geographically and topically) might energize some voters who wouldn't come out otherwise, we have enough barriers to voting other than enthusiasm, and enough state-level offices and issues bringing ot voters in "safe" states, that I think the direct impact of a popular vote on turnout will be low. Consider CSM's roundup of the states with turnouts >60% over the last six election cycles: Oregon, Alaska, South Dakota, Maine, Wisconsin, and Minnesota. Only one of those, Wisconsin, is a "battleground" state--CSM attributes most of the high turnout rate to voter-friendly laws like same-day registration or voting by mail and to traditions of vigorous local and state-level competitions, via state-level third party candidates or voter referenda.